*Estimating Physical Invariant Measures *

and Space Averages of Dynamical Systems Indicators

## Contents

### Part I: Lyapunov Exponents

#### Chapter 1: Estimation of Lyapunov Exponents of Dynamical Systems using a
Spatial Average

##### 1.1 Lyapunov Exponents of Deterministic Dynamical Systems

##### 1.2: Practical Estimation of Lyapunov Exponents

##### 1.3: Lyapunov Exponents of Random Dynamical Systems

##### 1.4: The Relationship between the Lyapunov Exponents of the Random and Deterministic
Systems

##### 1.5: Examples and Results

##### 1.6: Discussion

### Part II: Invariant Measures

#### Chapter 2: Computing Invariant Measures via Small Random Perturbations

##### 2.1: Introduction

##### 2.2: The Procedure

##### 2.3: Increasing the Accuracy of the Approximation

##### 2.4: Examples

##### 2.5: Does Any Small Random Perturbation Work?

##### 2.6: Discussion

#### Chapter 3: Finite Approximation of Sinai-Bowen-Ruelle Measures for Anosov
Systems in Two Dimensions

##### 3.1: Introduction

##### 3.2: Equilibrium States

##### 3.3: Approximation of the Weight Function

##### 3.4: Computing the Approximate Equilibrium State

##### 3.5: Discussion

#### Chapter 4: Approximating Physical Invariant Measures of Mixing Dynamical
Systems in Higher Dimensions

##### 4.1: Introduction

##### 4.2: Outline of Method

##### 4.3: *L^1 *and Strong Convergence

##### 4.4: Sensitivity of Finite Markov Chains

##### 4.5: Factors influencing the norm of the Fundamental Matrix

##### 4.6: The Behaviour of Mixing Constants for Two Classes of Maps, and Numerical
Results

##### 4.7: Discussion

#### Chapter 5: Mixing Properties and Aggregation

##### 5.1: Classes of Mixing

##### 5.2: Estimating the Rate of Mixing of Perturbed Systems

##### 5.3: Projected Perron-Frobenius Operators

##### 5.4: Aggregation

##### 5.5: Discussion

#### Summary